Characterization

To characterise each device as a PUF without directly calculating a bit output of the system, each peak position from each spectrum should be categorised by a region of the plane in which it occurs. The plane in which the peaks occur is divided up into uniform-width bins such that permutations can be compared. A method for this binning process is that any value falling within the bin is superseded by the median value of the bin. Hence, this allows the permutations peaks to be described by the bin in which it falls into.

The measurements here give a description of the entire system’s properties. These are denoted by robustness and uniqueness. Robustness is defined here as the similarity between repeat measurements of a single permutation of the system. Uniqueness is similarly defined as a measurement of how distinct the permutations of a system are

Figure 25: (a) 2D Gaussian Approximation to an ideal spread of data points across the Current voltage plane. Each axis split into 4, with 16 boxes covering the plane where points can be seen. Each point outputs a 4-bit signature. This can be increased/decreased based upon the spread of data. (b) 1D Gaussian approximation to spread of voltage shift of peak compared to original device. Axis split into 4, so each device outputs 2 bits, but can be varied based on data spread.

(a)

Chapter 3

52 Robustness and uniqueness can be calculated in a similar way and is based around what data points are compared. Upon comparing two data points a zero or one response is given based on if the data points are equal. The range of responses is binned by axis to create a way to categorise all peaks by where it occurs. A peak which lies within a bin is superseded by the median value of the bin such that all peaks which occur within a bin are equal. The value is compared to other measurements using equation 13 to calculate how distinct each permutation or subsequent measurement is. The distinction of permutations is shown by

𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) = {

0 𝑖𝑓 𝑅𝑖 = 𝑅𝑗

1 𝑖𝑓 𝑅𝑖 ≠ 𝑅𝑗

Where Ri̇= (Vi, Ii), denoting the voltage and current position of the ith iteration. (13)

Figure 26: Binning system for comparing peaks using equation 13 on how distinct each permutation is. Using equation 13, Green points would be equal and red point will be different.

The robustness of the system can be found for each peak of a permutation individually or more importantly, can be found for the permutation as a whole. Summing

𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) of subsequent measurements and dividing by the number of measurements taken would give the probability that a subsequent measurement will be different to the expected output. A measure of robustness would therefore be given by the inverse of the described function. This is denoted by the equation:

1 −1

k∑ 𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗,𝑡 ′ ₎ k

t=1

Where Ri and Ri,t gives the initial output and output at time t, respectively and k gives the number of total measurements. The ideal robustness for the given equation is 1, i.e. each subsequent measurement is guaranteed to have an output equal to that of the expected output and is reliable in its measurement.

The uniqueness is calculated in much the same way, relying on the distinctness of the expected output of a system. The chance that any two random inputs would produce an equivalent output can be found by calculating an average distinction across a set of permutations. A uniqueness measurement is calculated by taking 𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) of a permutation compared to all subsequent permutations. A sum of 𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) is multiplied by a coefficient which normalizes by the number of comparisons made. For the following equation:

2 k(k − 1)∑ ∑ 𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) k j=i+1 k−1 i=1

An ideal uniqueness for this is 1, where each permutation is distinct and has a unique output from all previous and subsequent measurements. The total number of permutations is given by k. Subsequent permutations are denoted with i and j.

(15) (14)

Chapter 3

54 Both uniqueness and robustness are used here to show the effectiveness of a system on providing reliable yet distinct responses from all of its possible permutations. The metrics of robustness and uniqueness correlate directly to the probabilities of false positives and false negatives. False positives are defined as the probability that a random incorrect challenge is accepted i.e when two responses to different challenges are too similar. False negatives are defined as the correct challenge is declined i.e. when subsequent measurements give a different response to the expected output. As robustness affects the minimum size of bins in the plane, to keep robustness high, means that it also affects the uniqueness of the responses. When bin size decreases, uniqueness increases whereas robustness decreases unless the system outputs a perfectly robust system. Hence, uniqueness and robustness need to be balanced to keep them both as high as possible as these values are connected to the probability of false positive and false negatives.

Uniqueness is difficult to calculate in terms of its relationship to subsequent permutations as it cannot be known if an output is truly unique or just deeply convoluted. For a true uniqueness to be calculated an infinite number of outputs would need to be tested and proven to be difficult or impossible to deconvolute. Devices can be fully tested to be entirely unique from one another, such that one device cannot be found from any of its predecessors, but would require and infinite number of measurements. However, an approximation can be found by using a large subset of the possible outputs and the previous equation. Hence in this research, a range of the different outputs are categorized to simulate an increasing subset which would tend towards showing that the system is truly unique for all permutations possible.

A further analytic measurement is given by Perm, which is a robustness of each permutation. This is defined by using each peak in a permutation in the same

𝑑𝑖𝑓𝑓(𝑅𝑖, 𝑅𝑗) measurement where one peak being different would constitute a 1 for that permutation measurement.

- Results and Discussion

A maximum of 16 similar 36µm2 _{resonant tunnelling diodes are used in this work to} create an exponential array representation. The arrays measured herein are created with the capacity to be characterised in a realistic time frame. Larger systems of arrays would have complete characterisation time which would render the need for protected access to be unnecessary. This property stems from the large set of CRPs and physical limitations present in a system and devices cause a single measurement time to be limited. The motivation being that a system can be challenged as many times in a reasonable time frame but still retain its security.

Responses are manipulated such that analysis can be performed to compare all permutations and peak positions. The peak outputs of each representation are normalised with respect to the point at which the tunnelling region of each constituent device begins. Therefore, using a linear approximation to the tunnelling gradient, extrapolation to zero current gives an approximation to where the peak would start from. Hence it is possible to shift the peak to a normalized position to compare a permutation peak to constituent devices.